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Practice Test: Number System- 3 - CAT MCQ


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20 Questions MCQ Test Quantitative Aptitude (Quant) - Practice Test: Number System- 3

Practice Test: Number System- 3 for CAT 2024 is part of Quantitative Aptitude (Quant) preparation. The Practice Test: Number System- 3 questions and answers have been prepared according to the CAT exam syllabus.The Practice Test: Number System- 3 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Practice Test: Number System- 3 below.
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Practice Test: Number System- 3 - Question 1

Raju had to divide 1080 by N, a two-digit number. Instead, he performed the division using M which is obtained by reversing the digits of N and ended up with a quotient which was 25 less than what he should have obtained otherwise. If 1080 is exactly divisible both by N and M, find the sum of the digits of N.

Detailed Solution for Practice Test: Number System- 3 - Question 1

According to the question, 

Given information is,Raju had to divide 1080 by N, a two-digit number. Instead, he performed the division using M, which is obtained by reversing the digits of N, and ended up with a quotient that was 25 less than what he should have obtained otherwise. If 1080 is exactly divisible both by N and M, find the sum of the digits of N.

1080 = 23 x 32 x 51.
Now, N x Q = 1080 and M x (Q - 25) = 1080
By Doing hit and trial method, 
N = 27
Hence the answer is 9.

Practice Test: Number System- 3 - Question 2

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

Detailed Solution for Practice Test: Number System- 3 - Question 2

Let the rectangle has x and y tiles along its length and breadth respectively. 
The number of white tiles 
W = 2x + 2(y – 2) = 2 (x + y – 2) 
And the number of red tiles = R = xy – 2 (x + y – 2) 
Given that the number of white tiles is the same as the number of red tiles 
⇒ 2 (x + y – 2) = xy – 2 (x + y – 2) 
⇒ 4 (x + y – 2) = xy 
⇒ xy – 4x – 4y = –8 
⇒ (x – 4) (y – 4) = 8 = 8 ×1 or 4 × 2 
⇒ m – 4 = 8 or 4 
⇒m = 12 or 8 
Therefore, the number of tiles along one edge of the floor can be 12

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Practice Test: Number System- 3 - Question 3

The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?

Detailed Solution for Practice Test: Number System- 3 - Question 3

There is only one number, 145, which exhibits this property.

Practice Test: Number System- 3 - Question 4

Let S be a two-digit number such that both S and S2 end with the same digit and none of the digits in S equals zero. When the digits of S are written in the reverse order, the square of the new number so obtained has the last digit as 6 and is less than 3000. How many values of S are possible?

Detailed Solution for Practice Test: Number System- 3 - Question 4

The correct option is Option A.

In this question, several restrictions are operating: If S and S2 are ending with the same unit digit, then it can be 0, 1,5,6, but it is given that none of the digits is equal to zero, so the unit digit can be only 1, 5, 6. Next, the unit digit of the square of the number written in reverse order is 6, so the tens place digit of the actual number should be either 4 or 6.

So, the actual numbers could be 41, 45, 46, 61, 65, 66.

Now, this square is less than 3000, so the only possibilities are 41, 45, 46.

Practice Test: Number System- 3 - Question 5

Let N be a positive integer not equal to 1. Then none of the numbers 2, 3,...., N is a divisor of (N! - 1). Thus, we can conclude that

Detailed Solution for Practice Test: Number System- 3 - Question 5

Eliminate the options.
For example, option (1) can be eliminated by assuming N = 2

Practice Test: Number System- 3 - Question 6

16 students were writing a test in a class. Rahul made 14 mistakes in the paper, which was the highest number of mistakes made by any student. Which of the following statements is definitely true?

Detailed Solution for Practice Test: Number System- 3 - Question 6

The number of mistakes made by all the students will be between 0 and 14, i.e., students are having a total of 15 options to make mistakes. Since the number of students = 16, at least two students will have the same number of mistakes (that can be zero also, i.e., two students are making no mistakes). Hence, option 1 is the answer.

Practice Test: Number System- 3 - Question 7

What is the remainder when (103 + 93)752 is divided by 123?

Detailed Solution for Practice Test: Number System- 3 - Question 7

A remainder can never be greater than the no. which is the dividing factor.
hence, the remainder < 123
which leaves us with only one option, i.e. (a)

Practice Test: Number System- 3 - Question 8

Every element of S1 is made greater than or equal to every element of S2 by adding to each element of S1 an integer x. Then, x cannot be less than: 

Detailed Solution for Practice Test: Number System- 3 - Question 8

Practice Test: Number System- 3 - Question 9

The History teacher was referring to a year in the 19th century. Rohan found an easy way to remember the year. He found that the number, when viewed in a mirror, increased 4.5 times. Which year was the teacher referring to?

Detailed Solution for Practice Test: Number System- 3 - Question 9

Going through the options, 8181/1818 = 4.5

Practice Test: Number System- 3 - Question 10

N is a number which when divided by 10 gives 9 as the remainder, when divided by 9 gives 8 as the remainder, when divided by 8 gives 7 as the remainder, when divided by 7 gives 6 as the remainder, when divided by 6 gives 5 as the remainder, when divided by 5 gives 4 as the remainder, when divided by 4 gives 3 as the remainder, when divided by 3 gives 2 as the remainder, when divided by 2 gives 1 as the remainder.What is N?

Detailed Solution for Practice Test: Number System- 3 - Question 10

Check it through the options.
Alternatively, answer will be LCM (2,3,4,..., 9,10) - 1 = 2520 - 1 = 2519

Practice Test: Number System- 3 - Question 11

How many different four digit numbers are there in the octal (Base 8) system, expressed in that system?

Detailed Solution for Practice Test: Number System- 3 - Question 11

The total number of numbers of four digits in octal system = 7 x 8 x 8 x 8 = 3584 When we convert this number into octal system, this is equal to 7000.

Practice Test: Number System- 3 - Question 12

A teacher wrote a number on the blackboard and the following observations were made by the students. The number is a four-digit number.The sum of the digits equals the product of the digits. The number is divisible by the sum of the digits.The sum of the digits of the number is

Detailed Solution for Practice Test: Number System- 3 - Question 12

Using options, the only possible value is 4112. The key here is: The sum of the digits equals the product of the digits.

Practice Test: Number System- 3 - Question 13

Find the unit digit:
346 765 * 768 983 * 987 599

Detailed Solution for Practice Test: Number System- 3 - Question 13

In this type of problem
Step 1: we find the unit digit of each term
Step 2: we find the product of the unit digits of each term
Step 3:  The unit digit of the product will be the product of whole number
The unit digit of 346 765 = 6
The unit digit of 768 983 = 2   (for unit digit  remainder of (power)/4 is checked and periodicity is checked as per base no ) like r(remainder)=983/4 is 3 so 83 unit digit is 2
The unit digit of 987599 = 3   (for unit digit  remainder of (power)/4 is checked and periodicity is checked as per base no ) like r(remainder)=983/4 is 3 so 73 unit digit is 3
6 * 2 * 3 = 36
Hence, the unit digit is 6.

Practice Test: Number System- 3 - Question 14

A certain number when successively divided by 4, 5 and 7 leaves remainders 2, 3 and 5 respectively. Find such a least number.

Detailed Solution for Practice Test: Number System- 3 - Question 14

To find the least number that, when successively divided by 4, 5, and 7, leaves remainders of 2, 3, and 5 respectively, follow these steps:

Step 1: Division by 4

Let the number be x. When x is divided by 4, it leaves a remainder of 2. This can be expressed as:

x = 4 * q1 + 2

Here, q1 is the quotient from this division.

Step 2: Division by 5

The quotient q1 from the first division is then divided by 5, leaving a remainder of 3. This can be written as:

q1 = 5 * q2 + 3

Substituting this back into the equation for x:

x = 4 * (5 * q2 + 3) + 2
x = 20 * q2 + 14

Step 3: Division by 7

Next, the quotient q2 is divided by 7, leaving a remainder of 5. This is expressed as:

q2 = 7 * q3 + 5

Substituting this into the equation for x:

x = 20 * (7 * q3 + 5) + 14 x = 140 * q3 + 114

Finding the Least Number

To find the smallest positive integer x, set q3 to 0:

x = 140 * 0 + 114
x = 114

Verification

  1. Divide 114 by 4:
    • 114 divided by 4 is 28 with a remainder of 2.
  2. Divide the quotient (28) by 5:
    • 28 divided by 5 is 5 with a remainder of 3.
  3. Divide the next quotient (5) by 7:
    • 5 divided by 7 is 0 with a remainder of 5.

All the conditions are satisfied with the number 114.

Answer: The least such number is 114

Practice Test: Number System- 3 - Question 15

A number when divided by 841 gives a remainder of 87. What will be the remainder when we divide the same number by 29?

Detailed Solution for Practice Test: Number System- 3 - Question 15

Let the number be N and its quotient be k.
Then the number N can be written in the form of:
N = 841k + 87
Now, we have to find out the what will be the remainder when it is divided by 29.
The number is (841k + 87)
Let’s divide it by 29
(841k + 87)/ 29 
841 and 87 both are completely divisible by 29.
Therefore, the remainder when the number N is divided by 29 is 0.

Practice Test: Number System- 3 - Question 16

A number when divided by 48 leaves a remainder of 31. Find the remainder if the same number is divided by 24

Detailed Solution for Practice Test: Number System- 3 - Question 16

Correct option is B

Let the number be N
Since the number when divided by 48 leaves a remainder of 31,  we have N=48(n)+31 where 'n' is the quotient got by dividing number by 48

We can write N=24×2×n+31 = 24×(2n)+24+7 = 24(2n+1)+7

So, when N is divided by 24, remainder left is 7

Practice Test: Number System- 3 - Question 17

If n2 = 123456787654321, what is n?

Detailed Solution for Practice Test: Number System- 3 - Question 17

Observe the pattern given below.
112 = 121
1112 = 12321
11112 = 1234321 and so on
so, 111111112 = 123456787654321

Practice Test: Number System- 3 - Question 18

A number when divided by 703 gives a remainder of 75. What will be the remainder when we divide the same number by 37?

Detailed Solution for Practice Test: Number System- 3 - Question 18

Let the number be N and its quotient be k.
Then the number N can be written in the form of:
N = 703k + 75
Now, we have to find out the what will be the remainder when it is divided by 37.
The number is (703k + 75)
Let’s divide it by 37
(703k + 75)/ 37
703 is divisible by 37 hence, remainder will be 0 whereas, 75 when divided by 37 leaves remainder 1.
Therefore, the remainder when the number N is divided by 37 will be (0 + 1) i.e. 1.

Practice Test: Number System- 3 - Question 19

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3, then the sum of the two numbers is

Detailed Solution for Practice Test: Number System- 3 - Question 19

Assume the numbers are a and b, ab = 616

Practice Test: Number System- 3 - Question 20

The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?

Detailed Solution for Practice Test: Number System- 3 - Question 20

 Let A = 100x + 10y + z  and B = 100z + 10y + x .According to given condition B - A = 99(z - x) As (B - A) is divisible by 7 . So clearly  (z - x) should be  divisible by 7.  z and x can have values 8,1 or 9,2 , such that 8-2=9 2=7 and  y can have  value from 0 to 9.
So Lowest possible value of A lowest x,y and z which is  is 108 and the highest possible value of A is 299.

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